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'''5767168/5767125''', the '''insanisma''', is a [[13-limit]] interval | '''5767168/5767125''', the '''insanisma''', is a [[13-limit]] interval. It is one of the smallest 13-limit [[comma]]s and the smallest 13-limit [[ultraparticular]], the difference between [[4096/4095]] and [[4225/4224]], or S64/S65 [[S-expression|S-expressed]]. | ||
As an ultraparticular, it is the difference between [[22/21]] and ([[65/64]])<sup>3</sup>. It also splits the [[olympia]] in three and the [[ibnsinma]] in two, each for 4096/4095~4225/4224. | As an ultraparticular, it is the difference between [[22/21]] and ([[65/64]])<sup>3</sup>. It also splits the [[olympia]] in three and the [[ibnsinma]] in two, each for 4096/4095~4225/4224. | ||
In terms of commas, it is also ([[1990656/1990625]])/([[123201/123200]]). | |||
== Temperaments == | == Temperaments == | ||
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[[Subgroup]]: 2.3.5.7.11.13 | [[Subgroup]]: 2.3.5.7.11.13 | ||
[[Comma list]]: | [[Comma list]]: 5767168/5767125 | ||
[[Mapping]]: | |||
{| class="right-all" | |||
|[⟨ | |||
|1 | |||
|0 | |||
|0 | |||
|0 | |||
|2 | |||
|7 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|1 | |||
|0 | |||
|0 | |||
|0 | |||
|0 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|1 | |||
|0 | |||
|0 | |||
| -1 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|0 | |||
|1 | |||
|1 | |||
|0 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|0 | |||
|0 | |||
|3 | |||
|1 | |||
|]] | |||
|} | |||
Mapping generators: ~2, ~3, ~5, ~7, ~65/128 | Mapping generators: ~2, ~3, ~5, ~7, ~65/128 | ||
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Badness (Sintel): 0.241 | Badness (Sintel): 0.241 | ||
For the [[19-limit]] weak extension, see [[insatinismic]]. | |||
==== 2.3.5.7.11.13.29 subgroup ==== | |||
[[Comma list]]: 5767168/5767125, [[176176/176175]] | |||
{| class="right-all" | |||
|[⟨ | |||
|1 | |||
|0 | |||
|0 | |||
|0 | |||
|2 | |||
|7 | |||
|15 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|1 | |||
|0 | |||
|0 | |||
|0 | |||
|0 | |||
| -3 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|1 | |||
|0 | |||
|0 | |||
| -1 | |||
| -3 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|0 | |||
|1 | |||
|1 | |||
|0 | |||
|3 | |||
|], | |||
|- | |||
|⟨ | |||
|0 | |||
|0 | |||
|0 | |||
|0 | |||
|3 | |||
|1 | |||
|7 | |||
|]] | |||
|}[[Optimal tuning|Optimal tunings]]: | |||
: [[WE]]: ~2 = 1199.999527¢, ~3/2 = 701.95558¢, ~5/4 = 386.314433¢, ~7/4 = 968.825693¢, ~65/128 = -1173.154534¢ | |||
: error map: -0.000473 0.000579 0.000719 -0.000214 -0.001218 0.000062 0.001014 | |||
Badness (Sintel): 0.211 | |||
== Etymology == | == Etymology == | ||